Ramification current, post-critical normality and stability of holomorphic endomorphisms of $\mathbb P^k$

TL;DR

This paper establishes equivalences between stability, post-critical normality, and ramification current convergence in holomorphic endomorphisms of projective space, simplifying previous approaches and clarifying their interrelations.

Contribution

It proves that stability is equivalent to a summability condition and ramification current convergence, improving understanding of stability in holomorphic dynamics.

Findings

Stability is equivalent to a summability condition for post-critical mass.

Stability corresponds to convergence of a ramification current.

The approach of [BBD18] is simplified and clarified.

Abstract

In the context of holomorphic families of endomorphisms of $\mathbb P^k$, we prove that stability in the sense of [BBD18] is equivalent to a summability condition for the post-critical mass and to the convergence of a suitably defined ramification current. This allows us to both simplify the approach of [BBD18] and better relate stability to post-critical normality.